SEVENTH EDITION and EXPANDED SEVENTH EDITION

Chapter 4 Systems of Numeration

Additive, Multiplicative, and Ciphered Systems of Numeration 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration

Systems of Numeration A system of numeration consists of a set of numerals and a scheme or rule for combining the numerals to represent numbers A number is a quantity. It answers the question “how many?” A numeral is a symbol used to represent the number (amount).

Types Of Numeration Systems Four types of systems used by different cultures will be discussed. They are: Additive (or repetitive) Multiplicative Ciphered Place-value

Additive Systems An additive system is one in which the number represented by a set of numerals is simply the sum of the values of the numerals. It is one of the oldest and most primitive types of systems. Examples: Egyptian hieroglyphics and Roman numerals.

Multiplicative Systems Multiplicative systems are more similar to the Hindu-Arabic system which we use today. Example: Chinese numerals.

Ciphered Systems In this system, there are numerals for numbers up to and including the base and for multiples of the base. The numbers (amounts) represented by a specific set of numerals is the sum of the values of the numerals.

Examples of Ciphered Systems: Ionic Greek system (developed about 3000 B.C. and used letters of Greek alphabet as numerals). Hebrew system Coptic system Hindu system Early Arabic systems

Place-Value or Positional-Value Numeration Systems 4.2 Place-Value or Positional-Value Numeration Systems

Place-Value System The value of the symbol depends on its position in the representation of the number. It is the most common type of numeration system in the world today. The most common place-value system is the Hindu-Arabic numeration system. This is used in the United States.

Place-Value System A true positional-value system requires a base and a set of symbols, including a symbol for zero and one for each counting number less than the base. The most common place-value system is the base 10 system. It is called the decimal number system.

Hindu-Arabic System Digits: In the Hindu-Arabic system, the digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Positions: In the Hindu-Arabic system, the positional values or place values are: … 105, 104, 103, 102, 101, 100.

Expanded Form To evaluate a number in this system, begin with the rightmost digit and multiply it by 1. Multiply the second digit from the right by base 10. Continue by taking the next digit to the left and multiplying by the next power of 10. In general, we multiply the digit n places from the right by 10n-1 in order to show expanded form.

Example: Expanded Form Write the Hindu-Arabic numeral in expanded form. a) 63 b) 3769 Solution: 63 = (6 x 101 ) + (3 x 1 ) or (6 x 10) + 3 3769 = (3 x 1000) + (7 x 100) + (6 x 10) + 9 or (3 x 103 ) + (7 x 102 ) + (6 x 101 ) + (9 x 1 )

4.3 Other Bases

Bases Any counting number greater than 1 may be used as a base for a positional-value numeration system. If a positional-value system has a base b, then its positional values will be … b4, b3, b2, b1, b0. Hindu-Arabic system uses base 10.

Example: Converting from Base 8 to Base 10 Convert 45368 to base 10. Solution:

Example: Converting from Base 5 to Base 10 Convert 425 to base 10. Solution:

Example: Convert to Base 3 Solution: The place values in the base 3 system are …, 36, 35, 34, 33, 32, 3, 1 or …729, 243, 81, 27, 9, 3, 1. The highest power of the base that is less than or equal to 342 is 243.

Example: Convert to Base 3 continued Successive division by the powers of the base gives the following result.

Example: Convert to Base 3 continued The remainder, 0, is less than the base, 3, so further division is necessary.

Computers Computers make use of three numeration systems Binary Octal Hexadecimal

Numeration Systems Binary system Octal system Hexadecimal system Base 2 It is very important because it is the international language of the computer. Computers use a two-digit “alphabet” that consists of numerals 0 and 1. Octal system Base 8 Hexadecimal system Base 16

Computation in Other Bases 4.4 Computation in Other Bases

Addition An addition table can be made for any base and it can be used to add in that base. Base 5 Addition Table 13 12 11 10 4 3 2 1 +

Example: Using the Base 5 Addition Table 235 Solution: From the table 45 + 35 = 125 Record the 2 and carry the 1. 1445 25

Example: Using the Base 5 Addition Table continued Add the numbers in the second column, (15 + 45) + 25 = 105 + 25 = 125. Record the 12. 1445 235 1225 The sum is 1225.

Subtraction Subtraction can also be performed in other bases. When you “borrow” you borrow the amount of the base given in the subtraction problem. Example: If you are subtracting in base 5, when you borrow, you borrow 5.

Multiplication Multiplication table for the given base is extremely helpful. Base 5 Multiplication Table 31 22 13 4 14 11 3 2 1 x

Example: Using the Base 5 Multiplication Table Multiply 125 x 35 Solution: Use the base 5 multiplication table to find the products. When the product consists of two digits, record the right digit and carry the left digit.

Example: Using the Base 5 Multiplication Table continued Record the 1 carry the 1. 125 x 35 1 (35 x 15) + 15 = 45 Record the 4. 415 The product is 415.

Division Division is carried out much the same way as long division in base 10. A division problem can be checked by multiplication. (quotient divisor) + remainder = dividend

Early Computation Methods 4.5 Early Computation Methods

Early Civilizations Early civilizations used a variety of methods for multiplication and division. Multiplication was performed by duplation and mediation, by the galley method, and by Napier rods.

Duplation and Mediation Duplation and mediation uses a pairing method for multiplication. Example: Multiply 13 22 using duplation and mediation. Solution: Write 13 and 22 with a dash to separate. Divide the number on the left in half, drop the remainder and place the quotient under the 13. Double the number on the right, and place it under the 22.

Duplation and Mediation continued 17 – 22 8 – 44 Continue this process until a 1 appears in the left hand column. 4 – 88 2 – 176 1 – 352 Cross out all the even numbers in the left-hand column and the corresponding numbers in the right-hand column. 17 – 22 8 – 44 4 – 88 2 – 176 1 – 352

Duplation and Mediation continued Now, add the remaining numbers in the right-hand column, obtaining 22 + 352 = 374. To check 17 22 = 374

The Galley Method The Galley method is also referred to as the Gelosia method. This method uses a rectangle split into columns and rows with each newly-formed square split in half by a diagonal.

Example: The Galley Method Multiply 426 65. Solution: Construct a rectangle consisting of 3 columns and 2 rows. Place the 3-digit number above the boxes and the 2-digit number on the right of the boxes. Place a diagonal in each box. Complete by multiplying.

Example: The Galley Method continued Add the numbers along the diagonals. The number is read down the left-handed column and along the bottom, as shown by the arrow. The answer is 27,690. 2 4 6 5 3 1 7 9

Napier Rods John Napier developed in the 17th century. Napier rods, proved to be one of the forerunners of the modern-day computer. Napier developed a system of separate rods numbered 0 through 9 and an additional strip for an index, numbered vertically 1 through 9. Each rod is divided into 10 blocks. Each block below contains a multiple of a the number in the first block, with a diagonal separating its digits. The units are placed to the right of the diagonals and the tens digits to the left.

Example: Napier Rods Multiply 6 284, using Napier rods. Solution: Line up the rods 2, 8, 4, using 6 as the index. To obtain the answer, add along the diagonals as in the galley method. Thus, 6 284 = 1704. 1 2 4 8 6 7